Computing Weakly Singular and Near-Singular Integrals Over Curved Boundary Elements
Pages A3728 - A3753
Abstract
We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the reference element's space using Newton's method, singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy of our method for quadratic basis functions and quadratic triangles with several numerical experiments, including the scattering by two half-spheres.
References
[1]
M. H. Aliabadi, W. S. Hall, and T. Phemister, Taylor expansions for singular kernels in the boundary element method, Internat. J. Numer. Methods Engrg., 21 (1985), pp. 221--2236.
[2]
B. K. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), pp. 1551--1584.
[3]
J.-J. Angélini, C. Soize, and P. Soudais, Hybrid numerical method for solving the harmonic Maxwell equations: I---Mathematical formulation, Rech. Aérospat. (English Ed.), 4 (1992), pp. 27--43.
[4]
J.-J. Angélini, C. Soize, and P. Soudais, Hybrid numerical method for solving the harmonic Maxwell equations: II---Construction of the numerical approximations, Rech. Aérospat. (English Ed.), 4 (1992), pp. 45--55.
[5]
J.-J. Angélini, C. Soize, and P. Soudais, Hybrid numerical method for solving the harmonic Maxwell equations: III---Iterative algorithm, code and validations, Rech. Aérospat. (English Ed.), 4 (1992), pp. 57--72.
[6]
O. P. Bruno and E. Garza, A Chebyshev-based rectangular-polar integral solver for scattering by geometries described by non-overlapping patches, J. Comput. Phys., 421 (2020), 109740.
[7]
O. P. Bruno and C. A. Geuzaine, An $\mathcal{O}(1)$ integration scheme for three-dimensional surface scattering problems, J. Comput. Appl. Math., 204 (2007), pp. 463--476.
[8]
O. P. Bruno and L. A. Kunyansky, A fast, high-order algorithm for the solution of surface scattering problems: Basic implementation, tests, and applications, J. Comput. Phys., 169 (2001), pp. 80--110.
[9]
X. Claeys and B. Delourme, High order asymptotics for wave propagation across thin periodic interfaces, Asymptot. Anal., 83 (2013), pp. 35--82.
[10]
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, SIAM, Philadelphia, 1983.
[11]
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Springer, New York, 2013.
[12]
B. Delourme, H. Haddar, and P. Joly, Approximate models for wave propagation across thin periodic interfaces, J. Math. Pures Appl., 98 (2012), pp. 28--71.
[13]
M. G. Duffy, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal., 19 (1982), pp. 1260--1262.
[14]
C. Geuzaine and J.-F. Remacle, Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), pp. 1309--1331.
[15]
R. D. Graglia and G. Lombardi, Machine precision evaluation of singular and nearly singular potential integrals by use of Gauss quadrature formulas for rational functions, IEEE Trans. Antennas and Propagation, 56 (2008), pp. 981--998.
[16]
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325--238.
[17]
M. Guiggiani and A. Gigante, A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, ASME J. Appl. Mech., 57 (1990), pp. 906--915.
[18]
M. Guiggiani, G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo, A general algorithm for the numerical solution of hypersingular boundary integral equations, ASME J. Appl. Mech., 59 (1992), pp. 603--614.
[19]
W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis, Springer, Berlin, 2015.
[20]
W. Hackbusch and S. A. Sauter, On the efficient use of the Galerkin-method to solve Fredholm integral equations, Appl. Math, 38 (1993), pp. 301--322.
[21]
W. Hackbusch and S. A. Sauter, On numerical cubatures of nearly singular surface integrals arising in BEM collocation, Computing, 52 (1994), pp. 139--159.
[22]
N. Hale and L. N. Trefethen, New quadrature formulas from conformal maps, SIAM J. Numer. Anal., 46 (2008), pp. 930--948.
[23]
W. S. Hall and T. T. Hibbs, Subtraction, expansion and regularising transformation methods for singular kernel integrations in elastostatics, Math. Comput. Model., 15 (1991), pp. 313--323.
[24]
S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, High-order Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane, Adv. Comput. Math., 40 (2014), pp. 245--272.
[25]
K. Hayami and C. A. Brebbia, A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements, in Mathematical and Computational Aspects. Boundary Elements IX, C. A. Brebbia, W. L. Wendland, and G. Kuhn, eds., Springer, Berlin, 1987, pp. 375--399.
[26]
S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles, IEEE Trans. Antennas and Propagation, 54 (2006), pp. 42--49.
[27]
B. Johnston, P. R. Johnston, and D. Elliott, A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method, J. Comput. Appl. Math., 245 (2013), pp. 148--161.
[28]
S. Kapur and V. Rokhlin, High-order corrected trapezoidal quadrature rules for singular functions, SIAM J. Numer. Anal., 34 (1997), pp. 1331--1356.
[29]
M. A. Khayat and D. R. Wilton, Numerical evaluation of singular and near-singular potential integrals, IEEE Trans. Antennas and Propagation, 53 (2005), pp. 3180--3190.
[30]
A. Klöckner, A. Barnett, L. Greengard, and M. O'Neil, Quadrature by expansion: A new method for the evaluation of layer potentials, J. Comput. Phys., 252 (2013), pp. 332--349.
[31]
R. Kress, Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Model., 15 (1991), pp. 229--243.
[32]
R. Kress, Linear Integral Equations, 3rd ed., Springer, New York, 2014.
[33]
M. Lenoir and N. Salles, Evaluation of 3D singular and nearly singular integrals in Galerkin BEM for thin layers, SIAM J. Sci. Comput., 34 (2012), pp. A3057--A3078.
[34]
F. M. Lether, Computation of double integrals over a triangle, J. Comput. Appl. Math., 2 (1976), pp. 219--224.
[35]
H. Ma and N. Kamiya, Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Eng. Anal. Bound. Elem., 26 (2002), pp. 329--339.
[36]
J. C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in $\mathbb{R}^3$, Comput. Methods Appl. Mech. Engrg., 8 (1976), pp. 61--80.
[37]
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006.
[38]
C. Pérez-Arancibia, C. Turc, and L. Faria, Planewave density interpolation methods for 3D Helmholtz boundary integral equations, SIAM J. Sci. Comput., 41 (2019), pp. A2088--A2116.
[39]
X. Qin, J. Zhang, X. Guizhong, Z. Fenglin, and L. Guanyao, A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element, J. Comput. Appl. Math., 235 (2011), pp. 4174--4186.
[40]
M. T. H. Reid, K. White, J, and S. G. Johnson, Generalized Taylor--Duffy method for efficient evaluation of Galerkin integrals in boundary-element method computations, IEEE Trans. Antennas and Propagation., 63 (2015), pp. 195--209.
[41]
D. Rosen and D. E. Cormack, Analysis and evaluation of singular integrals by the invariant imbedding approach, Internat. J. Numer. Methods Engrg., 35 (1992), pp. 563--587.
[42]
D. Rosen and D. E. Cormack, Singular and near singular integrals in the BEM: A global approach, SIAM J. Appl. Math., 53 (1993), pp. 340--357.
[43]
D. Rosen and D. E. Cormack, The continuation approach: A general framework for the analysis and evaluation of singular and near-singular integrals, SIAM J. Appl. Math., 55 (1995), pp. 723--762.
[44]
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869.
[45]
N. Salles, Calcul des singularités dans les méthodes d'équations intégrales variationnelles, Ph.D. thesis, Université Paris-Sud, 2013.
[46]
S. Sauter and C. Schwab, Boundary Element Methods, Springer, Berlin, 2011.
[47]
L. Scuderi, On the computation of nearly singular integrals in 3D BEM collocation, Internat. J. Numer. Meth. Engrg., 74 (2008), pp. 1733--1770.
[48]
T. W. Tee and L. N. Trefethen, A rational spectral collocation method with adaptively transformed Chebyshev grid points, SIAM J. Sci. Comput., 28 (2006), pp. 1798--1811.
[49]
J. C. F. Telles, A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Meth. Engrg., 24 (1987), pp. 959--973.
[50]
L. N. Trefethen, Approximation Theory and Approximation Practice, extended ed., SIAM, Philadelphia, 2019.
[51]
S. Vijayakumar and D. E. Cormack, An invariant imbedding method for singular integral evaluation on finite domains, SIAM J. Appl. Math., 48 (1988), pp. 1335--1349.
[52]
S. Vijayakumar and D. E. Cormack, A new concept in near-singular integral evaluation: The continuation approach, SIAM J. Appl. Math., 49 (1989), pp. 1285--1295.
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Published In
DOI:10.1137/sjoce3.44.6
Issue’s Table of Contents© 2022, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2022
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